ferent transcription factors (Wilkins, 1992; Davidson, 2001). Because

the control of most developmentally regulated genes is a consequence

of the synthesis of the factors that regulate their transcription, transi-

tions between cell types during development can be driven by changes

in the relative levels of a fairly small number of transcription factors

(Bateman, 1998). We can thus gain insight into the dynamical basis

of cell-type switching (i.e., differentiation) by focusing on molecular

circuits, or networks, consisting solely of transcription factors and

the genes that specify them. Such circuits, in which the components

mutually regulate one another™s expression, are termed autoregula-

tory. Cells make use of a variety of autoregulatory transcription-factor

circuits during development. It is obvious that such circuits, or net-

works, will have different properties depending on their particular

˜˜wiring diagrams,” that is, the patterns of interaction among their

components.

It is also important to recognize that transcription factors that

control cell differentiation not only initiate RNA synthesis on pro-

moters of template-competent genes (see above) but also are capable

of remodeling chromatin so as to transform their target genes from

the template-incompetent into the template-competent con¬guration

(Jimenez et al., 1992; Rupp et al., 2002).

Transcription factors can be classi¬ed as either activators, which

bind to a site on a gene™s promoter and enhance the rate of that

gene™s transcription over its basal rate, or repressors, which decrease

the rate of a gene™s transcription when bound to a site on its pro-

moter. The basal rate of transcription depends on constitutively active

transcription factors, which are distinct from those in the cell-type-

determining autoregulatory circuits that we will consider below. Re-

pression can be competitive or noncompetitive. In the ¬rst case, the

repressor will interfere with activator binding and can only depress

the gene™s transcription rate to the basal level. In the second case,

the repressor acts independently of any activator and can therefore

potentially depress the transcription rate below basal levels.

Keller (1995) used computational methods to investigate the be-

havior of several autoregulatory transcription-factor networks with a

range of wiring diagrams (Fig. 3.5). Each network was represented by

a set of n coupled differential equations -- one for the concentration

of each factor in the network -- and the steady-state behaviors of the

66 BIOLOGICAL PHYSICS OF THE DEVELOPING EMBRYO

XX XX

X

X

A B

XY

Y Y

C D

XY

XX

Y Y

E F

XX

X

Fig. 3.5 Six model genetic “circuits” encoding autoregulatory transcription factors.

(The diagrams represent circuits in the sense that there is feedback between their

components.) The green and blue horizontal bars represent the genes specifying the

transcription factors X (green symbols) and Y (blue symbols), respectively. The arrow

denotes the transcription start site. The portion of the gene to the left of the arrow is

the promoter. (A) Autoactivation by monomer X. (B) Autoactivation by dimer X2 . (C)

Mutual activation by monomer Y and dimer X2 . (D) Autoactivation by heterodimer XY.

(E) Mutual repression by monomers X and Y. (F) Mutual repression by dimer X2 and

heterodimer XY (with no binding site for the heterodimer on either promoter; see the

main text for details). Activating and repressing transcription factors are denoted by

circular and square symbols respectively.

systems were explored. The questions asked were the following. How

many stationary states exist? Are they stable or unstable?

To illustrate Keller™s approach, we discuss in more detail one such

network, designated as the ˜˜mutual repression by dimer and het-

erodimer” (MRDH) network, shown in Fig. 3.5F. It comprises the gene

encoding the transcriptional repressor X and the gene encoding the

protein Y, and thus represents a two-component network. Below we

summarize the salient points of the MRDH model.

1. The rate of synthesis of a transcription factor (i.e., d[X]/dt and

d[Y]/dt, [X] and [Y] being the respective concentrations of X and

Y) is proportional to the rate of transcription of the gene en-

coding that factor.

2. The transcription of a gene depends, in turn, on the spe-

ci¬c molecules that are bound to sites in the gene™s pro-

moter. Molecules can bind either as monomers (single protein

molecules, denoted by X, Y, etc.) or dimers (complexes of two

protein molecules). Both homodimeric complexes (X2 , Y2 ) and

heterodimeric complexes (XY) may be active. Thus a promoter

can be in various con¬gurations, with respective relative fre-

quencies, depending on its ˜˜occupancy,” that is, on how the

various potential factors bind to it. Speci¬cally, in the case of

the MRDH network, as characterized by Keller (Fig. 3.5F), the

3 CELL STATES: STABILITY, OSCILLATION, DIFFERENTIATION 67

promoter of gene X has no binding site for any molecule, either

activator or repressor: its only con¬guration is the empty state,

whose relative frequency therefore can be taken as 1. (Note that

the relative frequency of a promoter con¬guration is the prob-

ability of this con¬guration, thus the sum of all possible rela-

tive frequencies must be 1.) The (activator-independent or basal)

rate of synthesis of gene X is denoted by SXB . The promoter of

gene Y contains a single binding site for the dimeric form, X2 , of

the non-competitive repressor X. The monomeric forms of pro-

teins X and Y cannot bind DNA. Furthermore, proteins X and

Y can form a heterodimer XY that is also incapable of binding

DNA. Thus, while protein Y does not act directly as a transcrip-

tion factor, it affects transcription since it antagonizes the re-

pressor function of X by interfering with the formation of X2

(by forming XY). The promoter of gene Y can therefore be in two

con¬gurations: its binding site for X2 is either occupied or not

with respective relative frequencies K X K X2 [X]2 /(1 + K X K X2 [X]2 )

and 1/(1 + K X K X2 [X]2 ). (Note that the sum of these two relative

frequencies is 1.) Here K X and K X2 are the binding af¬nity of

X2 to the promoter of gene Y and the dimerization rate con-

stant for the formation of X2 , respectively, and we have used

the relationship K X [X2 ] = K X K X2 [X]2 . The synthesis of Y in both

con¬gurations is activator-independent and is denoted by SYB .

To incorporate the fact that X2 reduces the rate of transcription

of Y in a noncompetitive manner, in the occupied Y-promoter

con¬guration SYB is replaced by ρ SYB , with ρ ¤ 1.

3. The overall transcription rate of a gene is calculated as the sum

of products. Each term in the sum corresponds to a particular

promoter-occupancy con¬guration and is represented as a prod-

uct of two factors, namely, the frequency of that con¬guration

and the rate of synthesis resulting from that con¬guration. In

the MRDH network this rate for gene X is thus 1 — SXB (with SXB

as introduced above), because it has only a single promoter con-

¬guration. The promoter of gene Y can be in two con¬gurations

(with rates of synthesis SYB and ρ SYB , see above); therefore its

overall transcription rate is (1 + ρ K X K X2 [X]2 )SYB /(1 + K X K X2 [X]2 ).

4. The rate of decay of a transcription factor is a sum of terms,

each proportional to the concentration of a particular complex

in which the transcription factor participates. This is equiva-

lent to assuming exponential decay. For the transcriptional re-

pressor X in the MRDH network these complexes include the

monomer X, the homodimer X2 , and the heterodimer XY. If

the corresponding decay constants are denoted respectively by

dX , dX2 , and dXY then the overall decay rate of X is given by

dX [X] + 2dX2 K X2 [X]2 + dXY K XY [X][Y], K XY being the rate constant

for the formation of the heterodimer XY (i.e., [XY] = K XY [X][Y]).

The analogous quantity for protein Y is dY [Y] + dXY K XY [X][Y].

5. By de¬nition, the steady-state concentration of each transcrip-

tion factor is determined by the solution of the equation that

results from setting its rate of synthesis equal to its rate of

68 BIOLOGICAL PHYSICS OF THE DEVELOPING EMBRYO

(c)

1

[X] < 0

[Y] < 0

[Y]

[Yo ]

2

3

[Y] = 0

[X] > 0

[X] = 0

[Y] > 0

[X]

[Xo]

Fig. 3.6 The solutions of the steady-state Eqs. 3.3 and 3.4, given in terms of the

steady-state solutions [X0 ] and [Y0 ]. Here [X0 ] is de¬ned as the steady-state cellular

level of monomer X produced in the presence of a steady-state cellular level [Y0 ] of

monomer Y, assuming a rate of transcription SX0 . Thus, by de¬nition (see Eq. 3.3),

SX0 = d X [X0 ] + 2d X2 K X2 [X0 ]2 + d XY K XY [X0 ][Y0 ]. Since along the red and blue lines,

r r

respectively, d[X]/dt ≡ [X] = 0 and d[Y]/dt ≡ [ Y] = 0, it is the intersections of these

curves that correspond to the steady-state solutions of the system of equations 3.3 and

3.4. Steady states 1 and 3 are stable, whereas steady state 2 is unstable.

decay. With the above ingredients, for the MRDH network one

arrives at

d[X]

= SXB ’ {dX [X] + 2dX2 K X2 [X]2 + dXY K XY [X][Y]} = 0, (3.3)

dt

1 + ρ K X K X2 [X]2

d[Y]

= SYB ’ dY [Y] + dXY K XY [X][Y] = 0. (3.4)

1 + K X K X2 [X]2

dt

Keller found that if in the absence of the repressor X the rate

of synthesis of protein Y is high then in its presence the system de-

scribed by Eqs. 3.3 and 3.4 exhibits three steady states, as shown in

Fig. 3.6. Steady states 1 and 3 are stable and thus could be considered

as de¬ning two distinct cell types, while steady state 2 is unstable.

In an example using realistic kinetic constants, the steady-state val-

ues of [X] and [Y] at the two stable steady states differ substantially

from one another, showing that the dynamical properties of these

autoregulatory networks of transcription factors can provide the basis

for generating stable alternative cell fates during early development.

The validity of Keller™s approach depends on the assumption that

the steady-state levels of transcription factors determine cell-fate

choice. In most cases this appears to be a reasonable assumption (Bate-

man, 1998; Becker et al., 2002). Sometimes, however (e.g., at certain

stages of early sea urchin development), transitions between cell types

3 CELL STATES: STABILITY, OSCILLATION, DIFFERENTIATION 69

appear to depend on the initial rates of synthesis of transcription

factors and not on their steady-state values (Bolouri and Davidson,

2003). Under conditions in which Keller™s assumption holds, however,

one can ask whether switching between these alternative states is

predicted to occur under realistic biological conditions. This can be

answered af¬rmatively: the external microenvironment of a cell that

contains an autoregulatory network of transcription factors could

readily induce changes in the rate of synthesis of one or more of

the factors via signal-transduction pathways that originate outside

the cell (˜˜outside--in signaling,” Giancotti and Ruoslahti, 1999). The

microenvironment can also affect the activity of transcription factors

in an autoregulatory network by indirectly interfering with their nu-

clear localization (Morisco et al., 2001). In addition, cell division may

perturb the cellular levels of autoregulatory transcription factors, par-

ticularly if they or their mRNAs are unequally partitioned between

the daughter cells. Any jump in concentration of one or more fac-

tors in the autoregulatory system can bring it into a new basin of

attraction and thereby lead to a new stable cell state.

In several well-studied cases, transcription factors regulating dif-

ferentiation in a mutually antagonistic fashion are initially co-

expressed in progenitors before one is upregulated and others

downregulated. Using a framework similar to Keller™s, Cinquin and

Demongeot (2005) have explored how known interactions among such

factors may give rise to multistable systems for cell-type determina-

tion.

Dependence of differentiation on cell“cell interaction: the

Kaneko“Yomo “isologous diversi¬cation” model

We have seen that the existence of multiple steady states in a cleaving

egg makes it possible, in principle, for more than one cell type to arise

among the descendents of such cells. However, this capability does

not, by itself, provide the conditions under which such a potentially

divergent cell population would actually be produced and persist as

long as it is needed. In later chapters we will see how the early embryo

and various organ primordia use chemical gradients and other short-

and long-range signaling processes to ensure that the appropriate cell

types arise at the correct positions to make a functional structure. In

certain cases (possibly including the early mammalian embryo) the

¬rst step in the developmental process may be to generate a mixture

of different cell types in a relatively haphazard fashion, just to get

things going. As we will see in the next chapter, sometimes it does not

matter what the initial arrangement of cells is -- differential af¬nity

can cause cells to ˜˜sort out” into distinct layers and lead to organized

structures from a randomly mixed population.

Consider the following observations: (i) during muscle differenti-

ation in the early Xenopus embryo, the muscle precursor cells must